The Logic of Time Travel: Part I

  (Note added 8 Oct 2020: Diagrams have disappeared from this post and I do not know how to restore them.)

The basic concept of a time traveller is of someone who disappears and either reappears later or has appeared earlier. Disappearance is departure from the present. Later reappearance is arrival in the future. Earlier appearance would be arrival in the past. For story purposes, disappearance and (re)appearance are preferable to visible, tangible continuity between departure and arrival because they enable the traveller to arrive unexpectedly in the future or in the present when returning from the past. He also arrives either unaged and, in his own experience, instantaneously, or undergoes significantly reduced aging and duration. Otherwise he would merely endure time, like everyone else, not ‘travel through’ it.

Merely futurewards time travel would be merely invisible time dilation or temporal stasis. Therefore, time travel as such requires at least the possibility of pastwards travel, which includes returning to the present from the future. Anyone who arrives in the future but cannot return to the present is not a time traveller. Supernatural or magical time travel is fantasy whereas scientific or technological time travel is science fiction (sf).


Travel into the past reverses the order of cause and effect and therefore creates the causality paradoxes. When appearance or arrival precedes disappearance or departure, then an effect precedes its cause and might therefore cause it (this would be the circular causality paradox) or even prevent it (this would be the causality violation paradox). Causality violation is also called ‘changing the past’ because the cause can be prevented only by altering the course of preceding events. Altering events here means making them different from what they would have been if they had produced the cause and also different from the time traveller’s memories of them. In this sense, violating causality involves changing the past. Similarly, any changing of the past involves violating causality. For example, a time traveller who:

arrives in the 1930’s;
remembers having departed from the 1990’s with the intention of preventing the outbreak of World War II
and succeeds in preventing World War II;
thereby prevents his younger self from departing the 1990’s with the intention of preventing World War II.

In fact, in this case, he changes the course of so many lives that he almost certainly also prevents his own birth. Thus, his arrival in the 1930’s, and even his very existence, are, apparently, uncaused.


A paradox is either an actually or only an apparently self-contradictory proposition. Causality violation is on the face of it, actually contradictory: the cause occurs, because it produces its effect, and does not occur, because it is prevented by its effect.


There are several possible solutions, which are discussed further below. One involves divergent timelines. A timeline is a single, chronologically linear, sequence of events like familiar twentieth century history in which particular events occur on particular dates and only on those dates:


1914 beginning of World War I
1917 Russian Revolution
1918 end of World War I
1939 beginning of World War II
1945 end of World War II
1963 assassination of Kennedy
1969 first moon-landing
1991 end of Cold War, etc.


A sequence of events in which World War I occurred earlier or later or not at all or had a different outcome etc would be a different timeline. A sequence of events that was identical with our history until 1939 but then differed because, e.g., World War II did not occur could be described, from 1939 onwards, as a diverging timeline. Divergent timelines enter time travel fiction as a possible solution to the causality violation paradox because it can be argued that the cause occurs in an original timeline but is prevented in a divergent one. However, alternative and multiple timelines can also be considered in their own right simply as alternative versions of historical periods or of fictitious characters’ lives. (Stories simply about alternative timelines usually involve sideways travel between parallel universes in present time with no backwards or forwards time travel and thus are not time travel stories.) The questions of whether there can be more than one timeline and, if so, how they are related are considered further below while trying to solve the causality violation paradox.

Possible Solutions to the Causality Violation Paradox

1. The Logical Impossibility of Causality Violation in a Single Timeline
 Hitler was not killed or kidnapped in childhood. Therefore, either no-one tried to do this or someone tried and failed, even if we do not know the precise reason for failure: accident, apprehension, arrest etc and even if the person who tried had foreknowledge of Hitler’s career and had wanted to prevent it. This remains true even if the source of foreknowledge was time travel from Hitler’s future. In fact, introducing time travel increases the number of possible reasons for failure. It would not be easy for a time traveller from a later period to find the best time and place at which to intervene in order to interrupt Hitler’s early life, which might even be protected by pro-Nazi time travellers. Therefore, causality violation is logically impossible because it cannot be true that something happened when it is already known that it did not happen. More specifically, it cannot be true that someone performed a task when it is already known that it was not performed, thus that, if anyone tried, they failed.


Unfortunately, if time travel became common, then the number of adverse circumstances and accidents necessary to prevent causality violation would become statistically impossible. Therefore, there has to be some reason either why time travel does not occur (e.g. because it is impossible or because it is so difficult that no-one ever finds out how to do it in the whole history of the universe) or why its use is extremely limited (e.g. because it requires too much energy or because the theory is valid which stipulates that any time journey would also have to be a long space journey, thus precluding effortless disappearance and reappearance on Earth’s surface in recent historical periods).


(Added on 21 April 2006: Thanks to Colin G. Mackay for pointing out that there is no such thing as a "statistical impossibility". My argument should be re-phrased: Something must happen to prevent a logical contradiction but, by definition, it is extremely improbable that an extremely improbable event will always occur just in time to prevent any logical contradiction.)

2. The Immutability of Past Events

Past events, even when visited from the present, remain past and therefore cannot be affected by anyone from the present. Therefore, pastwards time travellers remain invisible, intangible and inaudible when in the past, not just when travelling to or through it. They are like ghosts from the future. Unfortunately, this is time-viewing, not time travel.

3. A Discontinuous Timeline

If causality is violated, then the prevented cause, e.g. the time traveller’s departure, simply does not occur, although it is ‘remembered’ by the appearing time traveller, and the effect, i.e. his appearance, is uncaused, therefore not really an effect. Events are discontinuous, as in quantum mechanics. This seems to be logically possible but is extremely paradoxical in the sense of apparently contradictory:

a traveller appearing now could be from the past, from the future or from no time;

 anyone leaving the present to kidnap Hitler will still fail because Hitler was not kidnapped but travellers from possible futures could prevent those futures;

some past events could have been caused by arbitrarily appearing time travellers that we do not know about;

 .c\xvvvba traveller arriving in what he regards as the past does not know whether he remembers a real future or a prevented one;

in fact, before leaving the present, a traveller knows that his memories are valid but, on arriving in the past, he does not know this!

Time travellers in this discontinuous timeline would almost certainly think in terms of divergent or multiple timelines even if time travel theory denied them.


The causal principle can be partly salvaged with the following diagram:

A

C

(B)

ABC is the usual causal order but here B is the time traveller’s departure and C is his arrival which prevents B. A is an earlier event, like the invention of a time machine, which would have caused B. Thus, A occurs and would have caused B which would have caused C which, however, occurs and prevents B. Thus, indirectly, A causes C. C would not have occurred without A although, to a casual observer, AC looks like a causality violation.

4. Divergent Timelines

If causality is violated, then a second timeline diverges from the original one:

Horizontal line =original timeline
Slanting line =divergent timeline
D =time traveller’s departure from his present
A =his arrival in the past
Questions:
Do the timelines diverge at the moment of the time traveller’s arrival?:

Or at some later moment when he definitely violates causality, e.g., by kidnapping Hitler?:

Sometimes, A would equal V:

e.g., if the time traveller appeared in front of the TV news reader the previous evening but, if he appeared unobserved in an otherwise empty room, would this change the past? Is the answer arbitrary: i.e., it would only change the past if we somehow knew that, in the original timeline, the room had remained empty? But why should the past not sometimes change without our knowledge?


In the scenario where the traveller can exist for a while in the past before violating causality:

two strange things happen at point V. First, to observers in the original timeline, the traveller simply disappears. Secondly, he creates around himself an entire new universe identical to the first except for any changes made by himself. How can this happen?

Having initiated and entered the new timeline, does he remain in it or revert to a moment just after point D on the original timeline? The latter does not seem to be logically necessary although I have seen it argued that the original timeline is the only one that he can properly exist in. What happens might partly depend on the method of time travel used. If the method involves sending a traveller into the past for a finite period, then recalling him to the present, the recalling to the present might or might not work between timelines, although I suspect that it would not.


If V does not prevent D from recurring on the divergent timeline, then what happens?

5. Disappearing Timelines

It is sometimes imagined that, if causality is violated, then the original timeline somehow ceases to exist but there are different possible meanings of this idea. One possible meaning is that in the original timeline, the entire universe ceases to exist at point D:

In this case, we would need to worry about anyone who intended to travel into the past in order to change history! However, although cessation of cosmic existence is logically possible at any time, it is not logically necessary in the case of causality violation, because the above discussion of diverging but co-existing timelines was not self-contradictory, and it is physically impossible according to empirically discerned conservation laws.


Further, if the entire universe were to cease to exist as a result of a causality violation, then it would more logically cease to exist at the moment of the violation, e.g., when Hitler is killed or kidnapped, not at any later moment, because, arguably, there would be no later moment in this timeline:

But then it makes more sense to say not that the original timeline has been caused to go out of existence but that it has been prevented from coming into existence. This implies a scenario not of two timelines, one of them ceasing to exist, but of one discontinuous timeline:

which was 3. above: A C (B)

6. Parallel Timelines

If we want more than one timeline but do not want time travellers creating universes, then we can say that the many universes simply co-exist but there are different ways to develop this idea. If the idea is used to explain causality violation by a time traveller, then its simplest version is as follows:

many initially identical timelines exist in parallel with each other;
whenever a time traveller moves either forwards or backwards in time, he also moves sideways into the next timeline;
therefore, when he seems to enter the past or future of his own timeline, he in fact enters the past or future of a second timeline identical to the first except for any changes that he makes in it, the first such change being his arrival:

(V may or may not prevent D in the second timeline but cannot affect D in the first timeline.)
There are three problems with this scenario. First, the time traveller does not get into the past of his own timeline, which is surely the purpose of time travel.

Secondly, because he does not get into his own past, there cannot be circular causality in this scenario: A in the second timeline cannot cause D in the first timeline. Of the two causality paradoxes, sf writers usually assume either that only circular causality is possible (because there is only one timeline) or that both are possible (presumably because timelines diverge when causality is violated but not otherwise) but not that only causality violation is possible (because they do not consistently assume the simplest parallel timelines theory). Thirdly, the time traveller never returns to his ‘present’ or starting point in the original timeline:

D2=departure from past for return to present
R=return to present, not until the third timeline

SF writers usually assume that, unless a time traveller changes the past, he can re-join his acquaintances in the original timeline, not just their duplicates in another timeline. The theory needs to be progressively complicated in order to allow for the kind of stories that get written.

First complication:

Futurewards time travel is along the same timeline and only pastwards travel is into a second timeline:

Second complication:

Only pastwards travel that leads to overt causality violation, like kidnapping Hitler, would take the time traveller into a second timeline. Thus, a mere reconnaissance into the past would look like this:

These two complications allow the time traveller to re-join his original acquaintances and allow A to cause D but also allow for causality violation.

A third complication would be that, after violating causality, the time traveller returns to the original timeline:

However, this complication is unnecessary because it is usually assumed that a traveller who changes history and returns to the present returns to an altered present which, in this scenario, is the present of the second timeline:

In this case, the traveller simply disappears from the point of view of his original acquaintances. Either they develop a parallel timelines theory or they conclude that attempted time travel is elaborate suicide.

Fourth possible complication:

Do two travellers from the same original timeline arrive in the same second timeline:

or not? Could there be a different series of timelines for each traveller, the second series at right angles to the page and others in further dimensions? This is another unnecessary complication which would prevent travellers from interacting with each other among the same historical events and would move the idea away from time travel towards parallel universes as such. Thus, parallel timelines with two out of four possible complications seems to be the simplest way to account for causality violation by time travellers.

7. Successive Timelines

There is a fifth possible complication of parallel timelines which allows for a third possible meaning of the idea that causality violation causes a timeline to cease to exist. In the parallel timelines theory as developed above, (i) the causality-violating time traveller enters each new timeline successively and (ii) the contents of each new timeline are determined by the contents of the previous timeline as modified by the time traveller. This suggests that there is a causal relationship between a timeline with a time traveller disappearing from it and the next timeline with the time traveller appearing in it. The phrases ‘new’, ‘successively’, ‘determined by’, ‘previous’, ‘causal relationship’ and ‘next’ suggest a temporal relationship between timelines.
We can imagine them not as paralleling each other in a fourth spatial dimension:

But as succeeding each other in a second temporal dimension:

If this is the case, then, from the point of view of a time traveller who has entered a new timeline:

his original timeline is now past in two senses. First, it is in his memory and thus in his subjective or personal remembered past. But in the discontinuous timeline scenario (3, above), this was compatible with the remembered timeline simply having no existence whatsoever. In the successive timelines scenario, the original timeline is in the objective past of the axis T2 and anything which is in an objective past can be said to have ceased to exist. In fact, the timeline will have ceased to exist from the point of view of a hypothetical four-dimensional observer for whom our T1 and T2 equal his S4 and T1, respectively.


This partly agrees with the way the issue is discussed in some sf stories. It is sometimes imagined that, if causality is violated, then the entire original timeline, from beginning to end, somehow ceases to exist. This is different from saying either that the universe ceases to exist at a particular moment of the original timeline or that the timeline has never existed. First, it existed. Now it does not. The terms ‘first’ and ‘now’ do not apply to any moments within the timeline itself. Subjectively, they apply to a temporal sequence in the time traveller’s memory.

Having ‘changed the past’, he experiences the ‘changed’ sequence of events while remembering an ‘original’ one. Objectively, the terms ‘first’ and ‘now’ apply either to nothing, as in the discontinuous timeline scenario, or (the only possible alternative) to a second temporal axis (T2) at right angles to the familiar direction of time (T1).


However, it is usually added in sf stories that cessation of a timeline means annihilation for its inhabitants so that, in their own self-interest, they should try to prevent D, the time traveller’s departure on his causality-violating mission. In fact, they would do better to prevent time travel or, failing this, to regulate it with a Time Patrol. Despite this assumption, it should be clear from the diagram that:

(i) T1 is the only arrow of time experienced by the inhabitants of the original timeline;
(ii) in T1, they exist until D and then continue to exist after it;
(iii) for them, the only result of D is that the time traveller disappears permanently from their universe;
(iv) in their experience, there is no moment at which their timeline ceases to exist;
(v) the time dimension in which the time traveller says that they have ceased to exist is not the one that their world lines extend along but is at right angles to it;
(vi) thus, all that they lose is some potential temporal width that they are not even aware of;
(vii) in the original timeline, an observer who knows about successive timelines can remark that this timeline is or will be (a new tense is needed) regarded as having ceased to exist by a time traveller;
(viii) however, this time traveller will himself have disappeared, or apparently ceased to exist, from the point of view of anyone who remains within the timeline.

Therefore, the inhabitants of the original timeline need not try to prevent D.

8. A Mutable Timeline

I have heard it argued that we can imagine one changeable timeline instead of many successive timelines. However, change involves a relationship between a state changed from and a state changed to. (If there were only one state, then there would be no change.) This relationship between changing states is a temporal relationship of ‘before’ and ‘after’.


Therefore, everything argued above about temporal relationships between timelines applies equally to temporal relationships between states of a timeline (if we want to use this changed terminology):


 

Inhabitants of an original state of the timeline need not be concerned that their state is or will be regarded as having ceased to exist by a time traveller who arrives in another state which has no existence on their T1 axis. The difference between successive timelines and a mutable timeline is merely terminological. Poul Anderson’s Time Patrollers sometimes speak as if they inhabit a single, discontinuous timeline, sometimes as if they inhabit a series of successive timelines and sometimes as if they inhabit a single mutable timeline. (1) (Although I have just argued that these last two options are identical, I here allow for characters and authors who seem to think otherwise.)


If they inhabit a single, discontinuous timeline, then some passages of some stories describe not events which did occur but events which would have occurred if they had not been prevented later in the story. We have to finish the story in order to be certain which events did occur. The passages set in ‘deleted timelines’ describe not experiences which the characters had but only experiences which they remember having had. The Patrollers definitely believe that any deleted timeline is utterly non-existent, even from its own point of view, i.e. that it does not have a point of view. This is consistent with the single, discontinuous timeline scenario.


On the other hand, even when they are in a particular timeline, Time Patrollers believe that it is possible that that timeline will be deleted and thus will turn out never to have existed in any sense whatsoever. I do not think that it is possible to make sense of this idea. First, if we assume a single, discontinuous timeline, then the deleted or prevented timeline should have no existence in the first place. Therefore the very fact that the Patrolmen are in the timeline while discussing it is already sufficient proof that it is not/has not been/will not be deleted.


Secondly, if we assume the parallel timelines/mutable timelines scenario(s), then yes, after causality has been violated, the original (state of the) timeline will have ceased to exist in the temporal dimension, T2, but it will still be true to say that it did exist in the past of T2. There will then be a subsequent (state of the) timeline. Within this timeline (I will now stop adding the qualifying phrase, ‘state of the…’), it will be true to say that the events of the original timeline have never occurred but this statement is true in T1, not in T2. It is not absolutely true. Patrolmen seem to be confusing questions of existence along two different temporal axes and their superhuman superiors, the Danellians, clearly have some reason to encourage this confusion.


However, the Patrolmen’s belief motivates them to preserve their timeline in T2 just as belief in God and the Devil motivated medieval monks to preserve Biblical and Classical literatures. Just as we can read about the Middle Ages without believing in the supernatural, so we can read about Time Patrol members without sharing their belief in the consequences of causality violation. A Time Patroller who is in his past has to prevent causality violations in order to be able to return to his version of the present and drama is added by his belief that that present would be not only inaccessible to him but also completely non-existent. The Time Patrol series more than compensates for any underlying incoherence by its colourful historical and even mythological reconstructions, its characterisation and its increasingly complicated time travel plots. Logical consistency is a major factor but not the only factor in the success or failure of time travel fiction.

9. Unwinding A Timeline

The Directors of Jack Finney’s top-secret government Project propose to send a time traveller into the past in order to alter Cuban history in the interests of U.S. foreign policy and they promise to be grateful to the traveller for completing this mission, clearly without considering that they and their memories will be affected by the alteration so that they will be unaware that it has happened or that they have benefited from it. (2) This may be an inconsistency, although a major one, in the minds of the characters rather than in the text of the novel. However, in the second novel, when history has been altered, a handful of people do have partial memories of other or ‘earlier’ timelines and even find physical evidence for them, like a ‘campaign button’ from John Kennedy’s second election campaign. (3) It is as if the causality violating time traveller had literally rewound a period of time and let it run forward again with an altered content but with some traces of the original content. This is presented as if it were an expectable, and not a highly questionable, consequence of causality violation. Four early chapters of the second Time novel serve only to move the action awkwardly, with the help of implausible sideways memories, from one timeline to another. (4) The characters could have learned that they were living in a subsequent timeline simply by reading the time traveller’s published account of the causality violation in which he prevented a meeting between the couple who would otherwise have been the parents of the founder of the Project. Then, without needing to have any memory of the Project, these characters could have set about retrieving it, as they do, by preventing the prevention of the meeting. However, Finney’s nostalgic accounts of past times and even of a lost timeline in which the Titanic did not sink and World War I was prevented more than compensate for any weakness in his time travel logic just as The Time Machine remains a classic despite the incoherence of its opening discussion of time first as a direction of static material extension and only of immaterial mental motion, then as a direction of variable material motion. (5) (In the latter case, there is nothing other than the entire universe for that universe to change its position in relation to, and the accelerating time machine should simply leave the rest of the universe behind it!)


To summarize theories of time travel:

1. The Logical Impossibility of Causality Violation in a Single Continuous Timeline
2. The Immutability of Past Events
3. A Single Discontinuous Timeline
4. Divergent Timelines
5. Disappearing Timelines
6. Parallel Timelines
7. Successive Timelines
8. A Mutable Timeline
9. Unwinding a Timeline

This is not necessarily a comprehensive list. A test of any story’s internal consistency could be: 

does it conform to any of these theories? If not, does it manage to be internally coherent by any other criteria? In practice, any given story will probably, if it is not completely incoherent, imply one of the above theories but not be fully consistent with it.

See Part II.

End-notes

1. Poul Anderson, The Time Patrol (New York: TOR Books, October 1991).
2. Jack Finney, Time and Again (New York: Simon & Schuster Inc, 1970; London: Arrow Books Ltd, 1990), pp. 386-387.
3. Jack Finney, From Time to Time (New York: Simon & Schuster Inc, 1996), pp. 20-21.
4. Finney, From Time to Time, pp. 42-65.
5. H. G. Wells, The Time Machine (Heinemann, 1895; London: Pan Books Ltd, 1953, 1973), pp. 7-12.